This article extends the definition of unbounded rational numbers established in my previous work, Every Real Number Can Be Expressed as an Unbounded Rational Number, to accommodate any chosen method for defining the integer part. While 𝜋 is typically defined as the ratio of a circle’s circumference to its diameter [1], it can also be expressed as an unbounded rational, inviting reconsideration of its traditional classification as irrational.
Other articles refuting Cantor’s Diagonal Argument
Cantor’s Diagonal Argument fails to pass the identity map sanity check
RCDA3: this article is the third installment of a series titled ’Refuting Cantor’s Diagonal Argument (RCDA)’ In RCDA1, I propose to use formal acceptance to Cantor’s Diagonal Argument,in the second I have applied a Double Cantor Diagonal Argument refuting Cantor’s Transfinite sets In this RCDA3 article I apply formal acceptance, as suggested in RCDA1, to perform a basic sanity check to Cantor’s Diagonal Argument using the identity map between the interval [0, 1) of real numbers and itself, CDA gives the usual non surjective results, which can not apply for bijective identity map. CDA’s failure to detect a bijection ensured by the identity map proves CDA is not a valid argument to qualify a one-to-one mapping between an arbitrary set E and the real numbers interval [0, 1).
There are obviously dramatic implications for transfinite numbers, which can not be discussed in this short article. . .
This work is licensed under CC BY-NC-SA 4.0
